4-dimensional Smale conjecture claims the following:

The inclusion $SO(5)$ → $SDiff(S^4)$ is a homotopy equivalence.

or Does $Diff(S^4)$ have the homotopy-type of $O(5)$ ?.

The inclusion $SO(n + 1$) → $SDiff(S^n)$ is a homotopy equivalence for n = 1 (trivial proof), n = 2 [1004,Smale,1959,Proc. Amer. Math. Soc.], n = 3 [464,Hatcher, 1983,Ann. of Math.], and is not a homotopy equivalence for n ≥ 5 [41,Antonelli, Burghelea, & Kahn,1972,Topology] and [164,Burghelea & Lashof,1974,Trans. Amer. Math. Soc.].

I looked everywhere but I could not find anything. Is this problem still open? Thanks.


This problem is completely open.


Tadayuki Watanabe has a preprint for the disproof here: https://arxiv.org/abs/1812.02448

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    $\begingroup$ A little more detail on Watanabe's paper: There is an equivalence $Diff(S^n) \simeq O_{n+1} \times Diff(D^n)$. Watanabe argues $\pi_i Diff(D^4)$ is non-trivial for $i \geq 1$. The $i=0$ case is closely connected to the 4-dimensional Schoenflies problem. At present it does not appear Watanabe's argument says anything about the $i=0$ case. $\endgroup$ – Ryan Budney Mar 12 at 19:44

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